It is often argued that a formulation of quantum mechanics in
terms of path integrals
is too advanced to lie within the scope of
most undergraduate courses. On the other hand, a great deal of
physics is done, nowadays, using Feynman's path integral method,
with applications ranging from gauge theories in high
energy physics to solid state
physics, and statistical mechanics. As a result, the amount of
pedagogical literature dedicated to a discussion of the path integral
approach to quantum mechanics is increasing steadily
[1,2,3,4,5].
Quantum mechanics is probably the most challenging paradigm of physics
that a student will ever come across, and the
pedagogical effectiveness of Feynman's method, beautifully expounded
by the author himself in his original work [6], lies in its
appeal to intuition while addressing the most fundamental principles
of the theory. Indeed,
it seems to us that one of the major achievements of Feynman's
formulation of quantum mechanics was to restore the particle trajectory
concept at the quantum level.
Our purpose, then, is to suggests a specific algorithm that emphasizes
the privileged role that classical trajectories play in
constructing the ``sum over histories'' envisaged by Feynman in order
to describe the quantum evolution of a particle.
This paper is an outgrowth of an earlier investigation in
string theory where the computational method described
here was first applied
[7,8,9]. However, in order to illustrate the method in
the simplest context, we apply it to the familiar case of a free, non
relativistic particle. We show that the path integral can be computed
exactly by summing over the whole family of trajectories which are
solutions of the classical equations of motion. This is in contrast
to the conventional approach to the path integral in which Newtonian
dynamics singles out a unique classical trajectory which is then
perturbed by quantum fluctuations. The main
ideas and techniques are discussed in the case of a free particle
with the bare
minimum of formal apparatus which should be within grasp of most
undergraduate physics majors taking a first course in quantum mechanics.
Hopefully,
the applicability of our approach to higher dimensional objects will
justify the claim, made in the abstract, that our method applies to
``simple quantum mechanical systems'' besides the point particle case
considered in this paper.
In order to place our work in the right perspective, we begin
Section 2 with a brief overview of the discretization
procedure that forms the basis of the ``sum over histories''
approach to Quantum Mechanics. This will give us the opportunity
to single out the precise point of departure of our own approach
from the conventional one based on the ``Lagrangian path integral''.
In Section 3 we construct the propagation kernel and make
contact with the diffusion equation of a free, non relativistic
particle.
In Appendix A we outline a possible extension
of our definition of path integral
to the case of a particle in an external potential.
Finally, Appendix B is a compendium of three basic
formulae which are especially relevant to the discussion in the text.
